Drawing Robot Pen Holders, Calligraphy Pens, and Thought Experiments

Just hanging out

Just hanging out

In discussing Sandy Noble’s Polargraph pen holder I mentioned how his design is optimized so that the point where the two cords meet is always the same as where the pen tip meets the paper.1 In the comments, he explained his rational, “So the pen tip is always at the tip of the hanging triangle, and there’s no distortion that way.”  My response was that “…if the pen holder has a single ‘hanging triangle’ point in it with the pen tip a constant distance from that ‘hanging triangle point,’ the drawings should appear identical to those created at the ‘hanging triangle point’ – just offset by the constant distance.”

This morning Sandy updated his site with a post explaining, using several diagrams, the basis for his prior theory and how he came to agree with my point. (Not that he’s changing his pen holders…  ;)  )

Without as much fancy-schmancy maths and geometry, I figured I would explain the thought experiment I used to conclude that a pen tip that is always a constant distance and position from the “hanging triangle point” will always produce an accurate distortion-free drawing.  To help illustrate these thought experiments, I’ve enlisted the help of Yoda.  “Hi Yoda!”

Fig 1:  Yoda, being drawn by a drawing robot

Fig 1: Yoda, being drawn by a drawing robot

In the picture above, Yoda is being drawn by a drawing robot.

Fig 2: Yoda drawing, annotated

Fig 2: Yoda drawing, annotated

Above, I’ve labeled the important parts of the drawing.  On the top left “Motor A,” on the top right “Motor B,” which are attached by cords to the pen holder indicated by the dark blue line.  Here, I’ve shown Yoda as he would be drawn by a drawing robot, where the robot then draws two more points.

Let’s say, because we’re feeling whimsical today, we want to add a second pen to our pen holder.  We’ll use a red pen and affix it below the blue pen in such a way that the red pen will always be directly below the blue pen by the same distance.

Fig 3:  Another pen

Fig 3: Another pen

For the moment let’s pretend the red pen is capped so it won’t leave a mark.  Now we’ll try to predict the position of the red pen at different points along the original drawing.

Fig 4: Where's the red pen?

Fig 4: Where’s the red pen?

It turns out this task is pretty easy.  The red pen, at any given point during the Yoda drawing, will always be directly below the blue pen by the same exact distance between the two pens.  Okay, now let’s draw Yoda again – this time with the red cap off.

Fig 5: Double vision

Fig 5: Double vision

We get two Yodas!  How awesome is that!  The reason I mentioned calligraphy pens in the title of this post is because it shows another way to think about this process.  When we write with a calligraphy pen we don’t have one end of the pen wildly distorted – in theory the two points on the calligraphy pen are always a constant distance from one another and moving together (as long as we don’t rotate the pen when we write).  You could imagine instead of a blue and red pen above, we’ve put a single calligraphy pen that’s as wide as the black line representing the distance between the two pens above.  The resulting drawing would look like a Yoda – that had been smudged downwards by the same distance.

Let’s now draw Yoda again, but capping the blue pen and still tracking where the blue pen would be.

Fig 6: Not using the blue pen

Fig 6: Not using the blue pen

We should end up with a result very similar to Fig 4.  It’s the same Yoda, only red and shifted down from the original blue Yoda by the distance between the two pens.

Let’s draw Yoda again – this time we’ve still got a pen holder which has the cord from Motor A meet Motor B at exactly one point.  As Sandy points out, this is really easy to do when you aren’t worrying about making that exact point be the same precise point as the pen tip.  Directly below point where the two cords meet on the pen holder, we’ll put the red pen.  From a functional standpoint, this setup is identical scenario to Fig 6.

Fig 7: Drawing just one red Yoda

Fig 7: Drawing just one red Yoda

Now we have a red Yoda, shifted down on the paper by the distance between the point where the two cords meet and where the red pen touches the paper.  It’s important to note that there’s no special magic to having the red pen directly below the point where the cords converge.  This pen tip just needs to be a constant distance and position from the cord convergence point at any given time.  While it might be more difficult to build a pen holder that holds the pen far off to one side, there’s no reason this wouldn’t work.

Fig 8: Yoda, now in green

Fig 8: Yoda, now in green

The lessons I take from this thought experiment are:

  1. As long as the pen is a constant distance and constant position from the point where the two cords meet, your drawing will not appear distorted – just shifted by the same constant distance and position.
  2. When calibrating the robot, the operator would need to calibrate the pen holder position by the cords convergence point – not the pen point.  This means that the preview in your software won’t match exactly the position of your drawing on the paper.  
  3. While not part of the thought experiment per se, I think we can all agree that the more weight that is not centered on the cord convergence point, the more likely the pen holder is to sway.
  4. I’m willing to defer to Sandy’s experience that pen holders that do not have the cord convergence point the same as the pen tip are, “Easier to design, easier to build, and cheaper, far, far cheaper.”

Thanks Yoda!

P.S.  Just in case you’re wondering – the reason that SVG of Yoda above is so large is because it includes the full TSP version of Yoda I’m getting ready to draw.  :)

”Posts
  1. DrawBot Resources and Links
  2. DrawBot, the Adventure Begins
  3. DrawBot - Parts Ordered!!!
  4. DrawBot - The Breakdown
  5. DrawBot - Parts Shipped!!!
  6. DrawBot - What would you draw?
  7. DrawBot - The Plan!
  8. DrawBot - The Hacks
  9. DrawBot – The Assembly, Part I
  10. DrawBot – The Assembly, Part III
  11. DrawBot – The Assembly, Part IV
  12. i find i want to add more posts in some random series, just so i can use my new plugin
  13. Simple Series with SEO! after just one day
  14. DrawBot – The Assembly, Part V
  15. DrawBot – The Assembly, Part VII
  16. DrawBot – The Operation, Part I
  17. DrawBot – The Assembly, Part VIII
  18. DrawBot – The Breakdown, Part II
  19. DrawBot – Printing!
  20. DrawBot – Calibration
  21. DrawBot – Pen Selection
  22. DrawBot – Pen Selection, Part II
  23. WordPress plugin - OCD Plugin Stats
  24. Restarting a Stalled DrawBot Drawing
  25. Another Drawing Robot!!!
  26. Why do DrawBots draw on walls?
  27. DrawBot Aesthetic Re-Design Ideas
  28. I think I know what I want to draw next...
  29. This project is not going to overengineer itself
  30. Overengineered Spools
  31. Overengineered Stepper Motor Mounts, Filament Guides
  32. Overengineered Bolt Endcaps, Case Holder
  33. Simple Series WordPress Plugin Update
  34. How to add a custom button to the WordPress Visual TinyMCE Editor
  35. A Study of Drawing Robot Pen Holders and Design Considerations
  36. Drawing Robot Pen Holders, Calligraphy Pens, and Thought Experiments
  37. Ideal Qualities in a Drawing Robot Pen Holder
  38. DrawBot Pen Holder Post Mortem
  39. To Maker Faire!!!
  40. Drawing Robot Penmanship

  1. Photo courtesy of Kristina Alexanderson []

4 Responses to “Drawing Robot Pen Holders, Calligraphy Pens, and Thought Experiments”

  1. […] the pen tip being at the same point as the cord convergence, the pen tip would be at some point a constant distance and position from the cord convergence point.  I think the reasons this type of holder isn’t seen is that it is so easy to build an […]

  2. Sandy Noble says:

    Good show, lesson 2 though – “When calibrating the robot, the operator would need to calibrate the pen holder position by the cords convergence point – not the pen point”. Isn’t there a difference between the geometric convergence point and the convergence point of the actual physical cords?

    With a gondola that has very widely offset attachment points (like your cardboard one), the actual physical convergence point would be much much lower than the actual gondola. I’m not sure where the geometric tip would be. I feel I know where it would be intuitively, but that doesn’t mean anything.

  3. MakerBlock says:

    @Sandy: “Isn’t there a difference between the geometric convergence point and the convergence point of the actual physical cords?” It depends. Using a Polargraph or Ragnar style pen holder, the geometric convergence point should be the same as the convergence point of the actual cords. This is also obviously the case where the cords are both tied to the same point, as in my pen holder.
    I agree that the geometric convergence point for widely offset attachment points could be lower than the pen tip, especially so in my crappy cardboard pen holder. Interestingly, I think the geometric convergence point would change slightly as the pen holder moved up and down.

  4. […] drawings on display back in 2013 – including a traveling salesman problem style Death Star, Yoda, and Nikola Tesla portrait.  This last one I gave to Joey Hurdy when he stopped by the booth just […]

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong>